Using a simple sketch state and proof the ParallelAxis Thêorem

The Parallel Axis Theorem is a fundamental principle in geometry and physics, particularly in the study of moments of inertia. Here's a step-by-step explanation and proof:

1. Statement of the Theorem: The Parallel Axis Theorem states that the moment of inertia of a body about an axis parallel to and a distance d away from an axis through its center of mass is equal to the moment of inertia of the body about the axis through its center of mass plus the product of its mass and the square of the distance between the axes.

Mathematically, this is expressed as: I = I_cm + md^2

Where:

• I is the moment of inertia about the parallel axis,
• I_cm is the moment of inertia about the center of mass axis,
• m is the mass of the body, and
• d is the distance between the two axes.

2. Sketch: Imagine a rigid body with a center of mass located at C. Draw two parallel axes: one (Axis 1) through the center of mass C, and another (Axis 2) a distance d away. The body is made up of many small particles, each with a mass mi and a distance ri from Axis 2.

3. Proof: The moment of inertia I about Axis 2 is the sum of mi*ri^2 for all particles. But each ri can be expressed as the sum of the distance di of the particle from the center of mass C and the distance d between the two axes. So, ri = di + d.

Substituting this into the expression for I gives: I = Σmi*(di + d)^2

Expanding this gives: I = Σmidi^2 + 2dΣmidi + d^2Σmi

The term Σmidi is zero because the center of mass is defined as the point where this sum is zero. So, the expression simplifies to: I = Σmidi^2 + d^2*Σmi

The first term on the right is the moment of inertia I_cm about the center of mass axis (Axis 1), and the second term is md^2. So, we have: I = I_cm + md^2

This is the Parallel Axis Theorem.